- #1

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## Homework Statement

[tex]F_{1}+F_{3}+F_{2n-1}[/tex]=[tex]F_{2n}[/tex]

## Homework Equations

## The Attempt at a Solution

P(k+1):[tex]F_{2k-1}+F_{2k+1}[/tex]=[tex]F_{2k+2}[/tex]

- Thread starter kathrynag
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- #1

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[tex]F_{1}+F_{3}+F_{2n-1}[/tex]=[tex]F_{2n}[/tex]

P(k+1):[tex]F_{2k-1}+F_{2k+1}[/tex]=[tex]F_{2k+2}[/tex]

- #2

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What happened to F_1 and F_3?

- #3

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well, it's F_1+F_3+....+F_2n-1What happened to F_1 and F_3?

- #4

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Then P(k + 1) needs to be changed accordingly.

What have you tried so far?

What have you tried so far?

- #5

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I've tried plugging in numbers.

- #6

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The definition of Fibonacci numbers will be helpful in the induction proof.

- #7

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If I plug in 1, I just get F_1, so 1=1

If I plug in 2, I get F_3, so 1+2=F_4, 3=3

If I plug in 2, I get F_3, so 1+2=F_4, 3=3

- #8

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F-1+F-3+...+F_2k-1+F_2k+1well, it's F_1+F_3+....+F_2n-1

P(k)+F_2k+1

F_2k+F_2k+1

Now I'm stumped...

- #9

HallsofIvy

Science Advisor

Homework Helper

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## Homework Statement

[tex]F_{1}+F_{3}+F_{2n-1}[/tex]=[tex]F_{2n}[/tex]

## Homework Equations

## The Attempt at a Solution

P(k+1):[tex]F_{2k-1}+F_{2k+1}[/tex]=[tex]F_{2k+2}[/tex]

well, it's F_1+F_3+....+F_2n-1

In your first post you said the problem was to prove that

[tex]F_{1}+F_{3}+F_{2n-1}[/tex]=[tex]F_{2n}[/tex]

Are you saying now it is actually to prove that

[tex]F_{1}+F_{3}+\cdot\cdot\cdot +F_{2n-1}[/tex]=[tex]F_{2n}[/tex]?

- #10

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Very close. What happens when 2 consecutive Fibonacci numbers are added?F-1+F-3+...+F_2k-1+F_2k+1

P(k)+F_2k+1

F_2k+F_2k+1

Now I'm stumped...

- #11

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It equals the 3rd Fibonnacci number.Very close. What happens when 2 consecutive Fibonacci numbers are added?

F_1+F-2=F_3

so F_2k+F_2k+1=F_2k+1+1

- #12

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And that's exactly what you wanted to show.

- #13

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Ok thanks!

- #14

epenguin

Homework Helper

Gold Member

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Then F

So if you are then asking also that

F

then

F

which is not making much sense.

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